Q.Show the circuit diagram by which we can obtain an output which is logarithm of input.

A logarithmic amplifier uses a diode (or transistor junction) in the feedback path of an inverting op-amp to exploit its exponential I–V law. …

[!TLDR]

A diode in the feedback path makes $V_o \propto -\ln V_{in}$.

Circuit: the input $V_{in}$ is applied through resistor $R$ to the inverting (−) input (virtual ground); a diode (or a transistor connected as a diode) forms the feedback path from output to that node; the (+) input is grounded.

Derivation: The diode obeys $I = I_s\left(e^{V/\eta V_T} - 1\right) \approx I_s e^{V/\eta V_T}$. At the virtual-ground node the input current equals the diode current:

Taking natural logarithms and rearranging,

$V_o = -\eta V_T \ln\!\left(\frac{V_{in}}{R I_s}\right)$ …